Method of an apparatus for remotely determining the profile of fluid turbulence

ABSTRACT

Apparatus and a method for remotely determining the profile of fluid turbulence such, for example, as clear-air turbulence. The intensity of an acoustic wave passing through a liquid, such as water, or the intensity of an electromagnetic wave, such as light, passing through air is determined at a plurality of different locations. In the electromagnetic case this may be done by a set of spaced telescopes and photosensitive devices or by optically scanning the density of a previously exposed photographic plate. An electric signal representative of the intensity of the acoustic or electromagnetic wave is then developed, and a second electrical signal is obtained which is representative of the spatial correlation function of the fluctuations of the logarithm of the first electrical signal. Thus, specifically the logarithm of the intensity is taken and the spatial correlation of the fluctuations is derived. Finally, a third signal is derived from the second signal. This third signal is representative of the integro-differential transform of the second signal. This third signal then represents the desired profile of, for example, the clear-air turbulence.

United States Patent [72] Inventors Saul Altshuler Manhattan Beach;Donald Arnush; Leonard Glatt, Palos Verdes; Arthur Peskotl", LosAngeles, Calif.

[21] Appl. No. 843,418

[22] Filed July 22, 1969 [45 Patented Mar. 2, 1971 [73] Assignee TRWInc.,

Redondo Beach, Calif.

[54] METHOD OF AND APPARATUS FOR REMOTELY DETERMINING THE PROFILE OFFLUID TURBULENCE 11 Claims, 10 Drawing Figs.

[52] U.S. Cl 235/l51.3,

[51] Int. Cl. G0lb 15/04 [50] Field ofSearch 356/105- [56] ReferencesCited UNITED STATES PATENTS 3,272,974 9/1966 MacCready, Jr. 235/151.3

3,360,793 12/1967 Collis 343/5 3,404,396 10/1968 Buehler et al.... 343/53,491,358 l/l970 Hicks 343/5 3,514,585 5/1970 Norsworthy 235/181 OPTICALSENSOR ARRAY (Z) 'r DISPLAY n DI ferenllctmg I Network OTHER REFERENCESProblems of Clear-Air Turblence: Possible Future Developments," E. R.Reiter; Astronautics & Aeronautics v.5 n8 Aug. 1967; pp.56- 58 PrimaryExaminerMalcolm A. Morrison Assistant Examiner-Edward J. Wise Att0rneysDaniel T. Anderson, Edwin A. Oser and Jerry A.

Dinardo ABSTRACT: Apparatus and a method for remotely determining theprofile of fluid turbulence such, for example, as clearair turbulence.The intensity of an acoustic wave passing through a liquid, such aswater, or the intensity of an electromagnetic wave, such as light,passing through air is determined at a plurality of different locations.In the electromagnetic case this may be done by a set of spacedtelescopes and photosensitive devices or by optically scanning thedensity of a previously exposed photographic plate. An electric signalrepresentative of the intensity of the acoustic or electromagnetic waveis then developed, and a second electrical signal is obtained which isrepresentative of the spatial correlation function of the fluctuationsof the logarithm of the first electrical signal. Thus, specifically thelogarithm of the intensity is taken and the spatial correlation of thefluctuations is derived. Finally, a third signal is derived from thesecond signal. This third signal is representative of theintegro-differential transform of the second signal. This third signalthen represents the desired profile of, for example, the clear-airturbulence.

Integrator PATENTED MAR 2 WI SHEET 2 BF 4 Fig- 3 Sutructor X p v) S 0ulAlfshuler onold Tlme Averoqer 437 Time Averuger Fig.4

42) Logarithmic Ampllfler Fig-6 ATTORNEY PATENTED "AR 2 |97| SHEET 3 OF4 [dz X Input 0 Integrator Multiplier Integrator Fig-8 Saul AlrshulerDonald Arnush Leonard Glo'rt Arthur Pa 8 koff INVENTORS ATTORNEYPATENTEU HAR 2 I97! sum u 0F 4 3,567,915

Soul Al'rshuler Donald Arnush Leonard Glut? Arthur Peskoff INVENTORS BYa (96h.

ATTORNEY BACKGROUND OF THE INVENTION This invention relates generally toa method of, and apparatus for remotely determining the profile ofturbulence in a fluid, and particularly relates to apparatus formeasuring or determining the clear-air turbulence profile.

Clear-air turbulence is widely recognized as a hazard to aircraftoperation. It is known that airplanes which have encountered suchclear-air turbulence have been broken up and crashed. This clear-airturbulence is sometimes due to a jetstream which may cause eddies toform as it passes, for example, over a mountain. However clear-airturbulence may exist near the ground at levels as low as-lO to 100meters. On the other hand, it has also been observed at heights up to50,000 or 60,000 feet. Clear-air turbulence has also been recognized asa significant meteorological parameter. In addition, it limits imagetransmission and laser communication through the atmosphere. I

Considerable effort has been expended by the aircraft industry andothers to combat the threat of clear-air turbulence. This howeverrequires initially some device to determine the profile of theturbulence of say, the atmosphere. What is really needed is aneffective, real-time, hazard warning device for airplanes which willrecognize the turbulence and determine its position: However at thistime there is no device known capable of determining the strength of theturbulence at remote locations.

It has been suggested to measure the fluctuations of images of stars. Itis, of course, well-known that these fluctuations are caused byclear-air turbulence. However up to now, there was no known method ordevice for determining the actual turbulence profile from these or othermeasurements.

It is also known that turbulence exists in the ocean. Since sea waterdoes not propagate electromagnetic waves well, it is more feasible tomeasure turbulence in the ocean by means of an acoustic wave.

It is accordingly an object of the present invention to provide a methodof, and apparatus for determining the profile of turbulence in either agaseous or liquid medium.

Another object of the present invention is to provide a method of, andapparatus for determining the profile of clearair turbulence bymeasuring the intensity of an electromagnetic or acoustic wave passingthrough the medium and subsequently processing the information obtainedat spaced locatrons.

A further object of the present invention is to provide apparatus forand a method of determining the location of clearair turbulence in theupper atmosphere.

SUMMARY OF THE INVENTION The mathematical theory upon which the presentinvention is based has been published by one of the inventors, ArthurPeskoff, in the Journal of the Optical Society of America, volume 58,no.8, pages 1032 to 1040 ofAug. 1968.

In any case, the apparatus of the present invention permits one todetermine remotely the profile of turbulence of a fluid such as water orair. This may be done in the case of water by an acoustic wave or in thecase of air by an electromagnetic wave passing through the fluid. Thuswhen it is desired to measure the clear-air turbulence, use may be madeof the light from a star passing through the atmosphere. Alternativelyother electromagnetic waves could be used such, for example, amicrowaves. These may be generated, for example, by an artificialsatellite passing over the atmosphere. Alternatively the artificialsatellite might illuminate the ground with a laser beam.

Thus the intensity of such a wave is first sensed at differentlocations. An electric signal is developed which represents theintensity of the wave. This may be effected by any suitable transducer.For example, in the case of clear-air turbulence a series of telescopesmay be set 'up, each provided with a photomultiplier or photodiode forconverting the light intensity into an electric signal representative ofdifferent locations. Alternatively a photographic plate may be exposedand developed and subsequently scanned by a photodensitometer to recordintensity variations as a function of location.

This first electric signal is then transformed into a second electricsignal by means of an analogue or digital computer.

The second electric signal is representative of the spatial cor-'relation function of the logarithm of the first electrical signal. Thecomputer then correlates the information obtained at the variouslocations where measurements were initially made. Finally additionalmeans are provided which may also consist of either an analogue ordigital computer. The third means serves the purpose to derive a thirdelectrical signal representative of the integro-differential transformof the second signal. Finally some display means such as a cathode-raytube may be provided for displaying the third electrical signal. Thisthird signal then represents the desired air-turbulence profile.

The novel features that are considered characteristic of this inventionare set forth with particularity in the appended claims. The inventionitself, however, both as to its organization and method of operation, aswell as additionalobjects and advantages thereof, will best beunderstood from the following description when read in connection withthe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS I at a set of concentric circles;

FIG. 4 is a schematic showing of a set of telescopes and associatedphotomultipliers for sampling light intensity at various locations;

FIG. 5 is a block diagram of one of the boxes of the system of FIG. 1;

FIG. 6 is a block diagram of another one of the boxes of the system ofFIG. 1;

FIG. 7 is a block diagram of the function compiler for correlating theinformation obtained from various spaced locations;

FIG. 8 is a block diagram of another'one of the boxes of FIG. I;

FIG. 9 is a chart of a typical logarithmic amplitude correlationfunction; and

FIG. 10 is a chart of a typical turbulence profile.

DESCRIPTION OF THE PREFERRED EMBODIMENTS Referring now to the drawingsand particularly to FIG. 1 there is illustrated in block form apparatusin accordance with the present invention. However, before describing theoperation of the block diagram of FIG. 1, it will be convenient to setforth the mathematical foundation on which the present invention isbased. Essentially it might be stated that any turbulence of a fluidsuch as clear air, creates random changes in the physical parameters ofthe air, such as the temperature, density and the index of refraction.The variation of the physical parameters of the medium in turn causesthe phase of a wave passing through the medium to change. The intensityvariations in the diffraction pattern caused by these phase phase of thewave also varies. After further propagation through the atmosphere,because of diffraction, the phase variations lead to intensityvariations.

The present invention is predicated on a mathematical equation for thestrength of the turbulence C,, (z). In this expression n stands for theindex of refraction of the fluid such as air, and 2 indicates thedistance from the observer measured along the line of sight.Accordingly, the following equation is obtained: I

0. (z =%fl (P, (P) P (I) wherein p is the separation between a pair ofsensors measuring the intensity of the electromagnetic or acoustic wave,and K(p, z) is the kernel of the transform given above by formula (IThis, for example, in the special case of atmospheric turbulence whichhas a so-called Kolmogorov spectrum, is given wsms s) In this equation F(x) is the well-known gamma function. Furthermore K is the wave numberof the electromagnetic wave. [m stands for the imaginary part of thequantity enclosed in curly brackets. Furthermore, F(a Iblw) is theconfluent hypergeometric function. Finally, B) (p) is the spatialcorrelation function of the fluctuations of the logarithmic amplitude ofthe wave. This function ,B (p) is defined as follows:

In this equation x is the logarithmic amplitude of the wave, that is,

X llog I, (4) where I is the intensity of the wave.

Angular brackets denote an average over time or space (in a planeperpendicular to the original propagation direction of the plane wave).It should be noted from equation (3) that the fluctuation of thelogarithmic amplitude of the wave is first determined. Furthermore, inthe equation 7indicates the coordinate in an XY system in theobservation plane (perpendicular to the original plane waves propagationdirection). Furthermore, the arrow over the letter indicates a vectorquantity.

Equation (1) is exact. However, as a practical matter, it would not bepossible to determine C,, (z) exactly because one cannot measure Bx(p)for values of p to infinity. In other words, the correlationmeasurements practically can only be carried out up to a certain finiteseparation, for example, of telescopes. Accordingly it may generally bemore convenient to use an approximate formula which gives a moreaccurate value for C i (2), when experimental errors are present in Bp). Accordingly the following approximate formula is obtained.

In the above equation the symbol ll indicates the approximate value ofA. Further, in Equation (6) p is an integration variable. p is a valueof p corresponding to the outer limit of correlation function which canbe measured. In other words, it is assumed that beyond the value p,,,there is too much noise to obtain a meaningful measurement. Thus thevalue of p which is dependent on the turbulence profile and theinstrumentation for measuring B)((p), is roughly that value of p forwhich the signal-to-noise ratio of Bx(p) is unity.

Having now laid the mathematical foundation of the present invention,reference is made to the block diagram of the system of the invention asshown in FIG. '1. The system includes a first box 10 which represents anoptical sensor array. Suitable arrays will be described subsequently inconnection with FIGS. 2, 3 and 4. In any case, such an optical sensorarray will deliver a plurality of electrical signals indicated as 1,, II each of which represents the intensity of the wave at a pan ticularlocation. There may be a total of N such sensors which generally will bemore than the three shown or possible less. Shown within dotted lines 11is an analogue or digital computer which computes the function Bx(p) ofequation (3). More specifically, the dotted line 11 includes a pluralityof boxes 12, l2,...., 12" which have been identified by A A ...,A,,,.The nature of each of boxes 12, etc., will be explained subsequently inconnection with FIG. 5. However, each of the boxes 12, etc., develops afunction x etc. This is an electrical signal representative of thefluctuations of the intensity at one particular location.

This signal is now fed into a next set of boxes 14, 14, 14" identifiedby B B B,,,. Each of the boxes 14, 14', develops a signal B (O), /3x(1), etc., corresponding to the correlation of the fluctuations of thelogarithm of the signal between a pair of sensors. The nature of each ofthe boxes l4, 14', etc., will be subsequently explained in connectionwith FIG. 6.

The electrical output signals of boxes 14, 14', etc., that is, Bx(0),Bx(l), etc., are fed into a function compiler 15. This may, for example,be considered to be in the nature of a timemultiplexer in which thevariable p may be converted to a time variable, and the general functionBx(p) is developed. This is the spatial correlation function of thepreviously calculated fluctuations of the logarithm of the initialelectrical signals 1,, etc. The nature of such a function compiler hasbeen shown by way of example in FIG. 7 to which reference will later bemade.

The thus obtained electrical signal corresponding to the function ,8x(p)is subsequently processed by the equipment shown within the dotted boxes16. This, in turn, performs the transformation of equations (5) (6) and(7). This, of course, yields the turbulence profile C,. (z) as theoutput of boxes 16 shown in FIG. 1. The final unit of the system of FIG.1 is a display unit 17, such, for example, as a cathode-ray tube or anyother convenient device for displaying or exhibiting the desiredfunction.

Within the dotted box 16 there is provided a multiplier 20 and analternative multiplier 21. As shown, the multiplier 20 will multiply thefunction Bx(p) with the function x(p,z) given in formula (2). By meansof the box 21, the )3 function is multiplied by L(p,z) in accordancewith equation (7). The result of the multiplication by units 20 or 21respectively, is B p)K(pz) or flx(p)L(p,z), as the case may be.

The multiplier 20 is then followed by an integrator 22, which performsthe integration shown, that is f dpBmKw) Similarly the multiplier 21 isfollowed by an integrator 23, which performs the following integrationThe integrator 23 is followed by a box 24 labeled C. The nature of thebox C has been shown in FIG. 8 to which reference will be madehereinafter. The output of the integrator 22 and of the box 24 is fedinto a summing network 25, which sums the two signals obtained from theintegrator 22 and the box 24. In other words, the results of theintegration, according to equation (5) of the integration in accordancewith equation (6) is summed by the summing network 25. The resulting sumis then differentiated again by a differentiating network 26, followingthe summing network 25. The result is the desired function C,, (z),which is fed into the display 17 as previously indicated.

It should be noted that multipliers such as shown at 20 and 21,integrators as shown at 22 and 23, as well as summing networks anddifferentiating networks such as boxes 25 and 26 are well-known in theart and need not be further described. Such operations could either beperformed by well-known analogue or digital computers, thus either aspecial purpose digital computer could be used or a suitably programmedgeneral purpose computer. The same applies to the equipment subsequentlydescribed in connection with FIGS. 5-9.

Referring now to FIG. 2, there is illustrated by way of example asuitable optical sensor array. This may, for example, consist of arelatively large converging lens 30 having a focal point 31 at the smallaperture in the otherwise opaque plane 29 which diverges the light, forexample, of a single star over a relatively large area. The light may bedetected in a plane indicated by the dotted lines 32. Each of the arrows33, 33', etc., may feed to a photoelectric device to provide the signalsI 1 etc., I

Alternatively there may be provided a photocell mosaic of the typeillustrated in FIG. 3. Thus there may be provided concentric rows ofphotocells or photocell multipliers as shown at 34, 35, 36 and 37. Eachphotoelectric device serves the purpose to measure the light intensityor the intensity of an electromagnetic wave at discrete locations and toprovide an electric output signal such as l,. 1

An alternative arrangement is shown in FIG. 4. This illustrates aplurality of telescopes 38, 38', 38", etc. These are disposed spacedfrom each other. For example, they may be disposed with a spacing of l,5, 3, 2 and 2 units, as will be subsequently explained in connectionwith FIG. 7. Each of the telescopes has associated therewith a photocellmultiplier such as shown schematically at 40, 40, 40", etc.

However, instead of detecting the intensity of the electromagnetic waveas shown in FIGS. 2 to 4, it may also be feasible to photograph, forexample, a star or a laser pulse, with the arrangement of FIG. 2 bypositioning a photographic plate in the plane 32. After the plate hasbeen developed and fixed it may be scanned by shining a light beamthrough it using, for example, a photodensitometer, and recording theresulting light intensity.

If it should be desired to measure the turbulence in the ocean, anacoustic wave may be generated in the ocean by a suitable transducer.This may, for example, be a loudspeaker or a piezoelectric crystal. Theintensity of the acoustic wave may again be measured with anothertransducer, such as an array of microphones.

As previously pointed out, the structure of the box 12 identified by A,is shown in FIG. 5. Accordingly, one of the electric signals, such asIYfeeds into a logarithmic amplifier 42. This will yield the signal YThe time-average of this signal is then obtained by the box 43 to obtainthe signal Y The subtractor 44 now subtracts the signal xYfrom thetimeaveraged signal xY to yield as shown Y Y This signal in turn feedsinto one ofthe boxes 14 or 14, etc. of FIG. 1.

The structure of these boxes has been shown by FIG. 6. Thus the twosignals uxu and Y Y feed into a multiplier 46. These two time-averagedand subtracted signals correspond to the initial signals obtained from apair of sensors. The multiplied signal is shown in FIG. 6 and thetimeaverage thereof is taken by the unit 47 to yield to signal Bx( xp.xY). This, of course, corresponds to the term Bxw), ,8x( 1 etc., asshown in FIG. 1 and these are the signals which feed into the functioncompiler 15.

This function compiler has been shown in FIG. 7 to which reference isnow made. As shown here, various signals indicated by 1,, 1 etc. are fedinto the boxes 12, 12, etc., identified by A A through A These signals1,, etc., correspond to the light intensities obtained from the opticalsensor array 10 arranged in such a way that the distances are as shownin FIG. 7, namely, respectively I, 5, 3, 2 and 2 units. The outputsignals of the boxes 12, 12, etc., are fed in such a way into the nextset of boxes 14, 14', etc. and identified by B B through B thataltogether l4 signals Bx(p,) are obtained in the manner shown. Forexample in order to obtain Bx(0) the box 14 is connected only to the box12, that is, only the light intensity I, is utilized. However, in orderto obtain the next signal B (1) the box 14' must be connected to bothboxes 12 and 12', that is, the light intensities I and 1 are utilized.The next signal is ,Bx(ll). This is obtained from the box 14''identified by B Its input is connected to the box identified A, and thebox identified by A Accordingly the signal corresponds to 1 1 units ofdistance which exist between the signals I and I composed of units 1, 5,3 and 2. It will be evident from the above explanation how the remainingsignals from Bx (0) through flx( 13) may be readily obtained.

It will be understood that FIG. 7 only shows by way of example how afunction compiler may be obtained. It will also be understood that therespective output signals ,Bx(0), etc. of FIG. 7 may be obtained eithersimultaneously in time or successively, that is, one after another.

Referring now to FIG. 8, there is illustrated the detailed structure ofthe box 24, identified by C of FIG. 1. This is one of the integraltransform subsystems. Its input is obtained from the integrator 23. Thefirst box 50 is another integrator performing the function indicated,that is, it integrates the input signal from zero to infinity obtainedfrom the integrator 23. The integrator 50 is followed by a multiplier51. This multiplies the output of the signal obtained from theintegrator 50 by (p/2) K (p,z), in the range pzp The output of themultiplier 51 is now once more integrated by the integrator 52 to obtainthe following signal f dpx input, where the input is the output of theoperation performed by the block 51. The output of the integrator 52then feeds into the summing network 25 of FIG. 1. The output of thesumming network 25 then feeds into the differentiating network 26 whichperforms the differentiation with respect to the variable z.

FIG. 9 to which reference is now made shows by way of example a curve55. This is a typical logarithic-amplitude correlation function andshows the function Bx(p) as a function of p. It will be noted that forlarge values of p the function [3 p) approaches zero in an oscillatorymanner.

A typical turbulence profile is shown by the curve 56 of FIG. 10.Accordingly the function C,, (z) is shown as a function of distance, z.Due to the increasing noise in the function B p) at increasingseparations p, it is not possible to obtain meaningful values for thisfunction for large values of p.

There has thus been disclosed apparatus and a method for the remotedetermination of the turbulence of air or liquid. The apparatus is basedon the recognition that such a turbulence will cause a random variationof various characteristics of the fluid, such, for example, as the indexof-refraction for an electromagnetic wave. As a result, the wave whichoriginally had a plane wave front becomes distorted and both intensityand phase of the wave are changedrin a manner which permitsdetermination of the profile of the turbulence. This is essentiallyeffected by initially measuring the intensity of the wave at differentlocations and then taking the spatial correlation function of thefluctuations of the logarithm of the light intensity. Finally theintegro-differential transform of the second signal is obtained toderive the desired turbulence profile which may then be displayed by asuitable display means.

We claim:

1. Apparatus for remotely determining the profile of turbulence of afluid by means of a periodic wave passing through the fluid comprising:

a. first means for sensing the intensity of the periodic wave atdifferent locations and for deriving a first electrical signalrepresentative of the periodic wave intensity at said differentlocations;

second means coupled to said first means for deriving a secondelectrical signal representative of the spatial correlation function ofthe fluctuations of the logarithm of said first electrical signal; and

c. third means coupled to said second means for deriving a thirdelectrical signal representative of the second signal, whereby saidthird signal represents the desired turbulence profile.

7 2. Apparatus for remotely determining the profile of clear- 7 fairturbulence by means of an electromagnetic wave passing through the aircomprising:

a. first means for sensing the intensity of the electromagnetic wave atdifferent locations and for deriving a first electrical signalrepresentative of the electromagnetic wave intensity at said differentlocations; second means coupled to said first means for deriving asecond electrical signal representative of the spatial correlationfunction of the fluctuations of the logarithm of said first electricalsignal;

c. third means coupled to said second means for deriving a thirdelectrical signal representative of the integro-differential transformof said second signal; and

. display means coupled to said third means for displaying said thirdsignal representing the desired air-turbulence profile.

3. Apparatus as defined in claim 2 wherein said first means includes aplurality of telescopes for sensing the intensity of the electromagneticwave in different locations, and a photoelectric device associated witheach of said telescopes for deriving said first electrical signal.

4. Apparatus as defined'in claim 2 wherein said first means includes alens for diverging the electromagnetic wave over an extended area withina predetermined plane, and a plurality of photoelectric devices disposedin said plane for deriving said first electrical signal.

5. Apparatus as defined in claim 2 wherein said second means includes alogarithmic amplifier for deriving a signal representative of thelogarithm of said first signal followed by a time-averager and asubtractor for obtaining the fluctuations of the logarithm of said firstelectrical signal.

6. Apparatus as defined inclaim 5 wherein said second means additionallyincludes a multiplier followed by an additional time-averager forderiving a plurality of signals, each being representative of thefluctuations of the logarithm of said first electrical signalcorresponding to a plurality of said different locations, and a functioncompiler coupled to said additional time-averagers for deriving acomposite signal representative of said spatial correlation function.

7. Apparatus as defined in claim 2 wherein said third means includes amultiplier, an integrator, a summing network and a differentiatingnetwork for deriving said third signal representative of theintegro-difierential transform of said second signal.

8. The method of remotely determining the profile of turbulence of fluidby means of a periodic wave passing through the fluid comprising thesteps of:

a. sensing the intensity of the periodic wave at a plurality ofdifferent locations;

b. deriving a first electrical signal representative of the periodicwave intensity at'the different locations;

c. deriving a second electrical signal from the first electrical signalrepresentative of the spatial correlation function of the fluctuations.of the logarithm of the first electrical signal; and

d. deriving a third electrical signal from the second electrical signalrepresentative of the integro-differential transform of the secondsignal, thethird signal representing the desired turbulence profile. 9.The method defined in claim 8 including the additional step ofdisplaying the third electrical signal to display the desired turbulenceprofile.

10. The method of remotely determining the profile of clear-airturbulence by means of an electromagnetic wave passing through the airand comprising the steps of:

a. sensing the intensity of the electromagnetic wave at differentlocations;

b. deriving a first electrical signal representative of theelectromagnetic wave intensity at the different locations;

0. deriving from the first electrical signal a second electrical signalrepresentative of the spatial correlation function of the fluctuationsof the logarithm of the first electrical signal;

(1. deriving from the second electrical signal a third electrical signalrepresentative of the'integro-differential transform of the secondsignal; and

e. displaying the third electrical signal for displaying the desiredclear-air turbulence profile.

11. A method as defined in claim. .0 including the additional steps ofderiving from the first electrical signal a first auxiliary signalrepresentative of the logarithm of the intensity of the electromagneticwave at the different locations, deriving from the first auxiliarysignal a second auxiliary signal representative of the fluctuations ofthe logarithm of the first auxiliary electrical signal, and derivingfrom the second auxiliary signal a third auxiliary signal representativeof the spatial correlation function.

1. Apparatus for remotely determining the profile of turbulence of afluid by means of a periodic wave passing through the fluid comprising:a. first means for sensing the intensity of the periodic wave atdifferent locations and for deriving a first electrical signalrepresentative of the periodic wave intensity at said differentlocations; b. second means coupled to said first means for deriving asecond electrical signal representative of the spatial correlationfunction of the fluctuations of the logarithm of said first electricalsignal; and c. third means coupled to said second means for deriving athird electrical signal representative of the second signal, wherebysaid third signal represents the desired turbulence profile. 2.Apparatus for remotely determining the profile of clear-air turbulenceby means of an electromagnetic wave passing through the air comprising:a. first means for sensing the intensity of the electromagnetic wave atdifferent locations and for deriving a first electrical signalrepresentative of the electromagnetic wave intensity at said differentlocations; b. second means coupled to said first means for deriving asecond electrical signal representative of the spatial correlationfunction of the fluctuations of the logarithm of said first electricalsignal; c. third means coupled to said second means for deriving a thirdelectrical signal representative of the integro-differential transformof said second signal; and d. display means coupled to said third meansfor displaying said third signal representing the desired air-turbulenceprofile.
 3. Apparatus as defined in claim 2 wherein said first meansincludes a plurality Of telescopes for sensing the intensity of theelectromagnetic wave in different locations, and a photoelectric deviceassociated with each of said telescopes for deriving said firstelectrical signal.
 4. Apparatus as defined in claim 2 wherein said firstmeans includes a lens for diverging the electromagnetic wave over anextended area within a predetermined plane, and a plurality ofphotoelectric devices disposed in said plane for deriving said firstelectrical signal.
 5. Apparatus as defined in claim 2 wherein saidsecond means includes a logarithmic amplifier for deriving a signalrepresentative of the logarithm of said first signal followed by atime-averager and a subtractor for obtaining the fluctuations of thelogarithm of said first electrical signal.
 6. Apparatus as defined inclaim 5 wherein said second means additionally includes a multiplierfollowed by an additional time-averager for deriving a plurality ofsignals, each being representative of the fluctuations of the logarithmof said first electrical signal corresponding to a plurality of saiddifferent locations, and a function compiler coupled to said additionaltime-averagers for deriving a composite signal representative of saidspatial correlation function.
 7. Apparatus as defined in claim 2 whereinsaid third means includes a multiplier, an integrator, a summing networkand a differentiating network for deriving said third signalrepresentative of the integro-differential transform of said secondsignal.
 8. The method of remotely determining the profile of turbulenceof fluid by means of a periodic wave passing through the fluidcomprising the steps of: a. sensing the intensity of the periodic waveat a plurality of different locations; b. deriving a first electricalsignal representative of the periodic wave intensity at the differentlocations; c. deriving a second electrical signal from the firstelectrical signal representative of the spatial correlation function ofthe fluctuations of the logarithm of the first electrical signal; and d.deriving a third electrical signal from the second electrical signalrepresentative of the integro-differential transform of the secondsignal, the third signal representing the desired turbulence profile. 9.The method defined in claim 8 including the additional step ofdisplaying the third electrical signal to display the desired turbulenceprofile.
 10. The method of remotely determining the profile of clear-airturbulence by means of an electromagnetic wave passing through the airand comprising the steps of: a. sensing the intensity of theelectromagnetic wave at different locations; b. deriving a firstelectrical signal representative of the electromagnetic wave intensityat the different locations; c. deriving from the first electrical signala second electrical signal representative of the spatial correlationfunction of the fluctuations of the logarithm of the first electricalsignal; d. deriving from the second electrical signal a third electricalsignal representative of the integro-differential transform of thesecond signal; and e. displaying the third electrical signal fordisplaying the desired clear-air turbulence profile.
 11. A method asdefined in claim 10 including the additional steps of deriving from thefirst electrical signal a first auxiliary signal representative of thelogarithm of the intensity of the electromagnetic wave at the differentlocations, deriving from the first auxiliary signal a second auxiliarysignal representative of the fluctuations of the logarithm of the firstauxiliary electrical signal, and deriving from the second auxiliarysignal a third auxiliary signal representative of the spatialcorrelation function.